3.4.92 \(\int \cot (e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [392]

Optimal. Leaf size=253 \[ -\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f} \]

[Out]

-2*arctanh((1+tan(f*x+e))^(1/2))/f-1/2*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+
2^(1/2))^(1/2)+1/2*ln(1+2^(1/2)+(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)-arcta
n(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f+arctan(((2+2*2^(1/2))
^(1/2)+2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f

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Rubi [A]
time = 0.19, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {3654, 12, 3566, 722, 1108, 648, 632, 210, 642, 3715, 65, 213} \begin {gather*} -\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}-\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {2 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]*(1 + Tan[e + f*x])^(3/2),x]

[Out]

-((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (
Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (2*Ar
cTanh[Sqrt[1 + Tan[e + f*x]]])/f - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x
]]]/(2*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(
2*Sqrt[1 + Sqrt[2]]*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3654

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan
[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2
, 0]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

\begin {align*} \int \cot (e+f x) (1+\tan (e+f x))^{3/2} \, dx &=\int \frac {2}{\sqrt {1+\tan (e+f x)}} \, dx+\int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=2 \int \frac {1}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {4 \text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}\\ &=-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}\\ &=-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}-\frac {\text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}\\ &=-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 78, normalized size = 0.31 \begin {gather*} \frac {-2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )+(1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+(1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]*(1 + Tan[e + f*x])^(3/2),x]

[Out]

(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2
)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/f

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.72, size = 2377, normalized size = 9.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2377\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)*(1+tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/8/f*((cos(f*x+e)+sin(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)+1)^2*(cos(f*x+e)-1)^2*(1+sin(f*x+e))*(16*I*2^(1/
2)*((sin(f*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x
+e))^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((sin(f*x+e
)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),-I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-1
6*I*2^(1/2)*((sin(f*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1
)/cos(f*x+e))^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((
sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^
(1/2))+3*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)
+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e
)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticE(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*(
(2-2^(1/2))/(2+2^(1/2)))^(1/2))-3*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((2^(1/2)*cos(
f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f
*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticF(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2
^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-6*EllipticE(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2)
)*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-s
in(f*x+e)+1)/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((sin(f
*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2)+16*EllipticF(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*
2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(
f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((sin(f*x+e)-1)*(1+2^(1/
2))/cos(f*x+e))^(1/2)-12*EllipticPi(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),2^(1/2)/
(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/
cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((sin(f*x+e)-1)*(1+2
^(1/2))/cos(f*x+e))^(1/2)+3*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)
*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-
2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticE(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*
2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-4*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*c
os(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*si
n(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticF(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2)
)*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^
(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(
f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticPi(1/2*((sin(f*x+e)-1)/cos
(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-4*((sin(f*x+
e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/co
s(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1
/2)*EllipticPi(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),-2^(1/2)/(2+2^(1/2)),I*((2-2^
(1/2))/(2+2^(1/2)))^(1/2))-12*EllipticE(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*((
2-2^(1/2))/(2+2^(1/2)))^(1/2))*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((2^(1
/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((sin(f*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2
)+12*EllipticF(1/2*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(
1/2))*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+co
s(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((sin(f*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2))*4^(1/2)/(cos(f*x+e)+si
n(f*x+e))/sin(f*x+e)^4/(2+2^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (202) = 404\).
time = 1.15, size = 878, normalized size = 3.47 \begin {gather*} -\frac {4 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right ) + 4 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right ) + 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} - 2 \, f\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 8^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} - 2 \, f\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) + 8 \, \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) - 8 \, \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right )}{8 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/8*(4*8^(1/4)*sqrt(2)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(-1/8*8^(3/4)*sqrt(2)*sqrt
(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + 1/8*8^(
3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 8^(1/4)*sqrt(2*s
qrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f*x + e) +
2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - sqrt(2)) + 4*8^(1/4)*sqrt(2
)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(-1/8*8^(3/4)*sqrt(2)*sqrt(2*sqrt(2)*f^2*sqrt(f^
(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + 1/8*8^(3/4)*sqrt(2*sqrt(2)*f^
2*sqrt(f^(-4)) + 4)*f^3*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) - 8^(1/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)
) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f*x + e) + 2*cos(f*x + e) + 2*sin
(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + f^2*sqrt(f^(-4)) + sqrt(2)) + 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4)) - 2*
f)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 8^(1/4
)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f
*x + e) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) - 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4)) - 2*f)*sqrt(2*sqr
t(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) - 8^(1/4)*sqrt(2*sqrt
(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f*x + e) + 2*c
os(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) + 8*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 1) - 8*
log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 1))/f

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot {\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(1+tan(f*x+e))**(3/2),x)

[Out]

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)*(1+tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e), x)

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Mupad [B]
time = 0.15, size = 85, normalized size = 0.34 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )}{f}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)*(tan(e + f*x) + 1)^(3/2),x)

[Out]

atan(f*((- 1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((- 1/2 - 1i/2)/f^2)^(1/2)*2i - (2*atanh((tan(e +
f*x) + 1)^(1/2)))/f - atan(f*((- 1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((- 1/2 + 1i/2)/f^2)^(1/2)*2
i

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